reserve x,y,X for set;

theorem Th21:
  for T being non empty TopSpace, A being Subset of T for x being
Point of T holds x in Cl A iff ex N being net of T st N is_eventually_in A & x
  is_a_cluster_point_of N
proof
  let T be non empty TopSpace, A be Subset of T;
  let x be Point of T;
  reconsider F = NeighborhoodSystem x as proper Filter of BoolePoset [#]T;
  hereby
    assume x in Cl A;
    then reconsider N = (a_net F)"A as subnet of a_net F by Th18,YELLOW_6:22;
    reconsider N9 = N as net of T;
    take N9;
    thus N9 is_eventually_in A by YELLOW_6:23;
    x is_a_convergence_point_of F, T by Th3;
    then
A1: x in Lim a_net F by Th17;
    Lim a_net F c= Lim N by YELLOW_6:32;
    hence x is_a_cluster_point_of N9 by A1,WAYBEL_9:29;
  end;
  given N being net of T such that
A2: N is_eventually_in A and
A3: x is_a_cluster_point_of N;
  consider i being Element of N such that
A4: for j being Element of N st i <= j holds N.j in A by A2;
  now
    let G be Subset of T;
    assume that
A5: G is open and
A6: x in G;
    Int G = G by A5,TOPS_1:23;
    then G is a_neighborhood of x by A6,CONNSP_2:def 1;
    then N is_often_in G by A3;
    then consider j being Element of N such that
A7: i <= j and
A8: N.j in G;
    N.j in A by A4,A7;
    hence A meets G by A8,XBOOLE_0:3;
  end;
  hence thesis by PRE_TOPC:def 7;
end;
